For the burgers: In a bowl, blend the ground beef with 1 teaspoon salt and 1/2 teaspoon pepper. Remove a small portion, about 1 ounce, and cook to check the level of seasoning. Add additional salt and pepper if needed.
Using about 8 ounces for each burger, form the meat into even-sized patties and flatten slightly (do not over-mix during the process).
Warm a griddle or pan over medium-high heat until hot, 2 to 3 minutes. Add the oil and patties and reduce the heat to medium. Cook on the first side until browned, 3 minutes, and then flip and repeat the process to the desired doneness. Remove and let rest.
For the pork: In a saucepan, warm the barbecue sauce over medium-low heat. Add the pulled pork and mix with the sauce until warm.
For the buns: In a flat pan, warm the buns over medium heat for 2 minutes. Butter the interior portion of the buns and cook until nicely toasted, 1 to 2 minutes.
To build the burgers, place the patties on the bottom buns. Top with the sauced pork and bun tops and serve.
Preheat the oven to 325 degrees F.
Sprinkle the Boston butt with the Cajun spice, salt and pepper and put in a deep roasting pan. Add the diced vegetables and bay leaves, and then cover with chicken stock. Wrap with plastic wrap and tightly cover with foil. Roast until tender, 2 to 2 1/2 hours.
Remove the pork from the stock and let cool. Shred with two forks.